Question

Evaluate $\mathbf{\nabla }\mathbf{.}{\mathit{e}}_{\mathbf{u}}\mathbf{,}\mathbf{\nabla }\mathbf{\times }{\mathit{e}}_{\mathbf{v}}\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\mathit{l}\mathit{n}\mathit{u}$

Short answer

The required values are given below.

$$\begin{gathered} \nabla ·{\hat{e}}_u=\frac{1}{u} \\ \nabla \times {\hat{e}}_v=\frac{\sqrt{2v-v^2}}{{\displaystyle \frac{\partial }{\partial u}}}\left(\frac{u}{\sqrt{2v-v^2}}.1\right){\hat{e}}_z=\frac{1}{u}{\hat{e}}_z \\ {\nabla }^2\ln u=0 \end{gathered}$$

Step-by-step solution

Given Information

The given values are mentioned below.

$$\begin{gathered} x=u\left(1-v\right) \\ y=u\sqrt{2v-v^2} \end{gathered}$$

Definition of a covariant vector

Components of a covariant vector or cotangent vector (commonly abbreviated as co vector) co-vary as the basis changes

Find the values

The scale factors are given below.

$$\begin{gathered} h_u=1 \\ {h}_v=\frac{u}{\sqrt{2v-v^2}} \end{gathered}$$

Find the value of$\nabla .{\hat{e}}_u$

$$\begin{gathered} \nabla .{\hat{e}}_u=\frac{1}{u}\frac{\partial }{\partial u}\left(u.1\right) \\ \nabla .{\hat{e}}_u=\frac{1}{u} \end{gathered}$$

Find the value of curl ${\hat{e}}_v$.

$$\begin{gathered} \nabla \times V=\frac{1}{h_u{h}_v}\left(\frac{\partial }{\partial u}\left(h_v{V}_v\right)-\frac{\partial }{\partial v}\left(h_u{V}_u\right)\right){\hat{e}}_z \\ \nabla \times {\hat{e}}_v=\frac{\sqrt{2v-v^2}}{{\displaystyle \frac{\partial }{\partial u}}}\left(\frac{u}{\sqrt{2v-v^2}}.1\right){\hat{e}}_z=\frac{1}{u}{\hat{e}}_z \end{gathered}$$

Find the value of Laplacian.

$\begin{array}{rcl}{\nabla }^{2}U& =& \frac{1}{h_u{h}_v}\left(\frac{\partial }{\partial u}\left(\frac{h_v}{h_u}\frac{\partial U}{\partial u}\right)+\frac{\partial }{\partial v}\left(\frac{h_u}{h_v}\frac{\partial U}{\partial v}\right)\right)\\ {\nabla }^2\ln u& =& \frac{1}{u}\frac{\partial }{\partial u}\left(u\frac{\partial }{\partial u}\ln u\right)\\ & =& 0\end{array}$

The required values are given below.

$\begin{array}{rcl}\nabla .{\hat{e}}_u& =& \frac{1}{u}\\ \nabla \times {\hat{e}}_u& =& \frac{\sqrt{2v-v^2}}{{\displaystyle \frac{\partial }{\partial u}}}\left(\frac{u}{\sqrt{2v-v^2}}\right){\hat{e}}_z=\frac{1}{u}{\hat{e}}_z\\ {\nabla }^{2}U& =& \frac{1}{h_u{h}_v}\left(\frac{\partial }{\partial u}\left(\frac{h_v}{h_u}\frac{\partial U}{\partial u}\right)\right)+\frac{\partial }{\partial v}\left(\frac{h_u}{h_v}\frac{\partial U}{\partial v}\right)\\ {\nabla }^2\ln u& =& 0\end{array}$